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Parameterization of Catmull-Clark Subdivision Surfaces

Parameterization of Catmull-Clark Subdivision Surfaces. CAI Hongjie | Nov 19, 2008. Catmull-Clark Subdivision Surface. parameterization. Motivations. Future GPU pipeline tessellation. Vertex Processing. Patch Assembly. Tessellation. Pixel Processing. Motivations.

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Parameterization of Catmull-Clark Subdivision Surfaces

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  1. Parameterization of Catmull-Clark Subdivision Surfaces CAI Hongjie | Nov 19, 2008

  2. Catmull-Clark Subdivision Surface parameterization

  3. Motivations • Future GPU pipeline • tessellation Vertex Processing Patch Assembly Tessellation Pixel Processing

  4. Motivations • Displacement Mapping Adaptive tessellation is needed to produce micropolygons

  5. Papers List • J. Stam. 1998 Exact Evaluation of Catmull-Clark Subdivision Surfaces at Arbitrary Parameter Values • C. Loop, S. Schaefer. 2008 Approximate Catmull-Clark Subdivision Surfaces with Bicubic Patches • H. Biermann, A. Levin, D. Zorin. 2000 Piecewise Smooth Subdivision Surfaces with Normal Control

  6. Origin of Catmull-Clark Surface • E. Catmull & J. Clark. 1978 Recursively Generated B-spline Surface on Arbitrary Topological Meshes • D. Doo. 1978 A Subdivision Algorithm for Smoothing down Irregularly Shaped Polyhedrons • D. Doo & M. Sabin. 1978 Analysis of the Behavior of Recursive Division Surfaces Near Extraordinary Points

  7. Knot Insertions of Uniform Bicubic B-Spline u5 u4 u6 u3 u5 u6 u7 u4 v6 v7 v6 v5 v5 v4 v4 v3

  8. Generating Control Points Face Point Edge Point Vertex Point

  9. Masks for New Points Face Point: Edge Point: Vertex Point: Vertex Point of valence n: cn bn bn cn cn an bn bn

  10. Continuity Around Extraordinary Point f2 e3 f1 e2 f3 v e1 e4 fn en

  11. Proceedings of SIGGRAPH 1998 Exact Evaluation of Catmull-Clark Subdivision Surfaces at Arbitrary Parameter Values Jos Stam

  12. Jos Stam • Curriculum vitae • 89-95 University of Toronto • 95-96 INRIA • 96-97 VTT • 97-06 Alias | Wavefront • 06-now Autodesk • Awards • SIGGRAPHComputer Graphics Achievement Award • Academy Award for Technical Achievement (2005) • Academy Award for Technical Achievement (2008)

  13. Neighborhood of Extraordinary Points

  14. Mathematical Setting

  15. Extended Subdivision Matrices

  16. Domain Branching Control Points Branching Surfaces Branching Domain

  17. Eigen Analysis Branching surfaces Eigen structure of subdivision surface Basis functions

  18. Illustrations of Basis Functions The last seven functions

  19. Results

  20. More Results

  21. ACM Transactions on Graphics,2008 Approximating Catmull-Clark Subdivision Surfaces with Bicubic Patches Charles Loop & Scott Schaefer

  22. Authors • Charles Loop • University of Utah, Master, 1987 “Smooth Subdivision Surfaces based on Triangles” • University of Washington, PhD,1992 • Microsoft Corporation • Scott Schaefer • Rice University, Master, 2003 • Rice University, PhD, 2006 • Texas A&M University, C.S. Assistant Professor

  23. Drawbacks of Stam’s Method • Can’t evaluate patches with more than one extraordinary points • One level subdivision is needed which increases memory and transfer bandwidth • Unfitted for animation

  24. Geometry Patches

  25. Exchange of Basis functions

  26. Masks for Bicubic Uniform B-Spline • Interior point: • Edge point: • Corner point:

  27. Generalized Masks • Interior point: • Edge point: • Corner point: 1 4 4 1 1 n2 4 4

  28. Tangent Patches As geometry patches can only meet with C0, so boundaries of tangent patches must be modified

  29. Masks for Tangent Patch Corners f0 e1 e0 fn-1 f1 u00 0 en-1 e2

  30. Tangent Patch Edges Purpose: G1 smooth along boundary curve u(t)

  31. Results

  32. More Results

  33. Comparisons with other Methods two subdivisions PN Triangles: Valchos et al.2001 PCCM: Peters, J. 2000

  34. More Comparisons one subdivision

  35. Proceedings of SIGGRAPH 2000 Piecewise Smooth Subdivision Surfaces with Normal Control Henning Biermann Adi Levin Denis Zorin New York University Tel Aviv University New York University

  36. Problems of Previous Subdivision Rules • No suitable C1 rules for boundary • Folds may result on concave corners

  37. Eigen analysis f2 e3 f1 e2 f3 v e1 e4 fn en

  38. Tagged Meshes • Edge tags: edges are tagged as crease edges to control the behavior of meshes • Vertex tags • Crease vertex • Corner vertex • Dart vertex • Sector tags • Convex sector • Concave sector

  39. Subdivision Rules • Masks for vertex points • Untagged and dart vertices: standard • Crease vertex: 1/8 —— 3/4 ——1/8 • Corner vertex: interpolated • Masks for face points: standard bn bn cn cn an bn bn

  40. Subdivision Rules • Masks for Edge point • Untagged edge: standard • Crease edge: midpoint • One end tagged edge:

  41. Flatness and Normal Modification • Flatness modification for concave corners • Normal modification

  42. Results

  43. Normal Interpolation

  44. More Results

  45. Thanks!

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